Your data matches 45 different statistics following compositions of up to 3 maps.
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Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000147
Mp00061: Permutations to increasing treeBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00204: Permutations LLPSInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [2]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 3 = 2 + 1
[7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 1 + 1
[6,7,5,4,8,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[6,5,7,8,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[7,8,6,3,4,5,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2 + 1
[7,8,5,3,4,6,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2 + 1
[7,8,3,4,5,6,2,1] => [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 3 + 1
[7,4,5,6,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[6,4,5,7,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[5,4,6,7,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[6,7,5,8,4,2,3,1] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 1 + 1
[8,7,4,5,6,2,3,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[8,6,4,5,7,2,3,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[7,6,4,5,8,2,3,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,7,5,8,3,2,4,1] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 1 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,5,6,2,3,4,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[8,6,5,7,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,6,5,8,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,5,8,2,3,4,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,6,3,4,2,5,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 1 + 1
[7,8,6,3,2,4,5,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,6,2,3,4,5,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,8,5,3,4,2,6,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 1 + 1
[7,8,4,3,5,2,6,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 1 + 1
[7,8,4,3,2,5,6,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,3,4,2,5,6,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,4,2,3,5,6,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,8,3,2,4,5,6,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[8,6,3,4,5,2,7,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[8,4,3,5,6,2,7,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[8,5,3,4,2,6,7,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,4,5,3,2,8,1] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 1 + 1
[7,6,3,4,5,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[7,5,3,4,6,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,5,3,4,7,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,4,3,5,7,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,7,5,4,2,3,8,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[7,6,4,5,2,3,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,4,5,2,3,8,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,5,4,7,2,3,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[5,6,4,7,2,3,8,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,5,3,2,4,8,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,4,3,2,5,8,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,3,2,4,5,8,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,4,3,5,2,6,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,5,3,4,2,7,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[5,4,3,6,2,7,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[4,5,3,2,6,7,8,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[8,6,7,4,5,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 1 + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00061: Permutations to increasing treeBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 77% values known / values provided: 77%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => [2]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 3 = 2 + 1
[7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 1 + 1
[6,7,5,4,8,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[6,5,7,8,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[7,8,6,3,4,5,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2 + 1
[7,8,5,3,4,6,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2 + 1
[7,8,3,4,5,6,2,1] => [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 3 + 1
[7,4,5,6,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[6,4,5,7,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[5,4,6,7,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[6,7,5,8,4,2,3,1] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 1 + 1
[8,7,4,5,6,2,3,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[8,6,4,5,7,2,3,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[7,6,4,5,8,2,3,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,7,5,8,3,2,4,1] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 1 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,5,6,2,3,4,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[8,6,5,7,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,6,5,8,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,5,8,2,3,4,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,6,3,4,2,5,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 1 + 1
[7,8,6,3,2,4,5,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,6,2,3,4,5,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,8,5,3,4,2,6,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 1 + 1
[7,8,4,3,5,2,6,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 1 + 1
[7,8,4,3,2,5,6,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,3,4,2,5,6,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[7,8,4,2,3,5,6,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,8,3,2,4,5,6,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[8,6,3,4,5,2,7,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[8,4,3,5,6,2,7,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[8,5,3,4,2,6,7,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,4,5,3,2,8,1] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 1 + 1
[7,6,3,4,5,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[7,5,3,4,6,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,5,3,4,7,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,4,3,5,7,2,8,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 + 1
[6,7,5,4,2,3,8,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[7,6,4,5,2,3,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,4,5,2,3,8,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,5,4,7,2,3,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[5,6,4,7,2,3,8,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,5,3,2,4,8,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,4,3,2,5,8,1] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2 + 1
[6,7,3,2,4,5,8,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,4,3,5,2,6,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[6,5,3,4,2,7,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[5,4,3,6,2,7,8,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 + 1
[4,5,3,2,6,7,8,1] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3 + 1
[7,8,5,6,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [2,1,4,3,5,6,8,7] => ?
=> ? = 1 + 1
Description
The length of the partition.
Matching statistic: St001039
Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St001039: Dyck paths ⟶ ℤResult quality: 64% values known / values provided: 64%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[8,7,6,5,3,4,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,7,6,4,3,5,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[8,7,5,4,3,6,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,7,5,3,4,6,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[8,7,4,3,5,6,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[8,6,5,4,3,7,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,6,4,3,5,7,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,6,5,3,4,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[7,6,4,3,5,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[7,5,4,3,6,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[6,5,4,3,7,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,5,6,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,6,5,7,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,5,8,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,5,3,2,4,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,5,6,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,5,8,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,5,2,3,4,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,6,5,7,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[7,6,5,8,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,7,6,4,3,2,5,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,3,2,4,5,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,7,6,2,3,4,5,1] => [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,7,5,3,4,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,4,3,5,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,5,4,2,3,6,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,7,5,3,2,4,6,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,7,4,3,2,5,6,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,6,5,4,3,2,7,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,5,4,6,3,2,7,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00121: Dyck paths Cori-Le Borgne involutionDyck paths
St000013: Dyck paths ⟶ ℤResult quality: 53% values known / values provided: 53%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,7,5,8,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,5,7,8,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[7,8,6,3,4,5,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[8,6,7,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[7,6,8,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[8,7,5,3,4,6,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[7,8,5,3,4,6,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[8,7,4,3,5,6,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[8,6,4,3,5,7,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,7,4,5,3,8,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,4,5,6,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,4,5,7,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[5,4,6,7,3,8,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[7,6,5,3,4,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[7,6,4,3,5,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[7,5,4,3,6,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[6,5,4,3,7,8,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[8,7,5,6,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,5,7,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,5,8,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,4,5,6,2,3,1] => [[[[.,[.,.]],[.,[.,.]]],[.,.]],.]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[8,5,6,4,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,4,5,6,7,2,3,1] => [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[7,5,6,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,5,7,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,7,4,5,8,2,3,1] => [[[[.,[.,.]],[.,[.,.]]],[.,.]],.]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[8,7,5,6,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,5,8,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,7,5,2,3,4,1] => [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,5,2,3,4,1] => [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,6,5,7,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,6,5,8,2,3,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[6,5,7,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,6,3,4,2,5,1] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000097
Mp00061: Permutations to increasing treeBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
St000097: Graphs ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,1,4,5,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,4,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,4,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [2,1,6,5,7,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,4,7,5,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,4,7,6,5] => [[.,[.,.]],[.,[[[.,.],.],.]]]
=> [2,1,5,6,7,4,3] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,5,4,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,5,4,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,5,6,4,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,5,6,7,4] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [2,1,6,5,4,7,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,5,7,4,6] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,5,7,6,4] => [[.,[.,.]],[[.,[[.,.],.]],.]]
=> [2,1,5,6,4,7,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,6,4,5,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,6,4,7,5] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,6,5,4,7] => [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [2,1,4,5,7,6,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,6,5,7,4] => [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [2,1,4,6,5,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,6,7,4,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,6,7,5,4] => [[.,[.,.]],[[[.,[.,.]],.],.]]
=> [2,1,5,4,6,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,7,4,5,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,1,7,4,6,5] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,7,5,4,6] => [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [2,1,4,5,7,6,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,7,5,6,4] => [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [2,1,4,6,5,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,7,6,4,5] => [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [2,1,4,5,7,6,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,1,7,6,5,4] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,3,4,1,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,1,5,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,4,1,6,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,4,1,6,7,5] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,4,1,7,5,6] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,4,1,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 2 + 1
[2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,6,1,5,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,6,1,7,5] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,6,5,1,7] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [4,5,3,2,1,7,6] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,7,1,5,6] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,7,1,6,5] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,7,5,1,6] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [4,5,3,2,1,7,6] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,4,7,6,1,5] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [4,5,3,2,1,7,6] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,5,1,4,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,3,5,1,4,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,5,1,6,4,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,5,1,6,7,4] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,5,1,7,4,6] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,5,1,7,6,4] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 2 + 1
[2,3,5,4,1,6,7] => [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [3,4,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,3,5,4,1,7,6] => [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [3,4,2,1,6,7,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 2 + 1
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000730
Mp00061: Permutations to increasing treeBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00240: Permutations weak exceedance partitionSet partitions
St000730: Set partitions ⟶ ℤResult quality: 27% values known / values provided: 27%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => {{1,2}}
=> 1
[2,1] => [[.,.],.]
=> [1,2] => {{1},{2}}
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ?
=> ? = 1
[7,8,5,6,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ?
=> ? = 1
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ?
=> ? = 1
[8,5,6,7,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ?
=> ? = 2
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ?
=> ? = 1
[6,7,5,8,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ?
=> ? = 1
[7,5,6,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ?
=> ? = 2
[6,5,7,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ?
=> ? = 2
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[8,7,4,5,6,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ?
=> ? = 2
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[8,6,4,5,7,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ?
=> ? = 2
[8,4,5,6,7,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ?
=> ? = 3
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[6,7,5,4,8,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 1
[7,5,6,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[6,5,7,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[7,6,4,5,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ?
=> ? = 2
[6,7,4,5,8,3,2,1] => [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [2,1,5,4,3,6,7,8] => ?
=> ? = 2
[7,5,4,6,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ?
=> ? = 2
[7,4,5,6,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ?
=> ? = 3
[6,5,4,7,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ?
=> ? = 2
[5,6,4,7,8,3,2,1] => [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [2,1,5,4,3,6,7,8] => ?
=> ? = 2
[6,4,5,7,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ?
=> ? = 3
[5,4,6,7,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ?
=> ? = 3
[8,7,6,5,3,4,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,6,5,7,8] => ?
=> ? = 1
[7,8,6,5,3,4,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [2,1,3,4,6,5,7,8] => ?
=> ? = 1
[8,6,7,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ?
=> ? = 1
[7,6,8,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ?
=> ? = 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[7,8,5,6,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [2,1,4,3,6,5,7,8] => ?
=> ? = 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[6,7,5,8,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [2,1,4,3,6,5,7,8] => ?
=> ? = 1
[6,5,7,8,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2
[8,7,6,4,3,5,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,6,5,7,8] => ?
=> ? = 1
[7,8,6,4,3,5,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [2,1,3,4,6,5,7,8] => ?
=> ? = 1
[7,6,8,4,3,5,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ?
=> ? = 1
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,2,3,6,5,4,7,8] => ?
=> ? = 2
[7,8,6,3,4,5,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2
[8,6,7,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,3,2,6,5,4,7,8] => ?
=> ? = 2
[7,6,8,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,3,2,6,5,4,7,8] => ?
=> ? = 2
[8,7,5,4,3,6,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,6,5,7,8] => ?
=> ? = 1
Description
The maximal arc length of a set partition. The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
Matching statistic: St001203
Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00030: Dyck paths zeta mapDyck paths
St001203: Dyck paths ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[8,5,6,7,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[6,7,5,8,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,5,6,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[6,5,7,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,4,5,6,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,6,4,5,7,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[8,4,5,6,7,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[6,7,5,4,8,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,5,7,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,6,4,5,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[6,7,4,5,8,3,2,1] => [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[7,5,4,6,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[7,4,5,6,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[6,5,4,7,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[5,6,4,7,8,3,2,1] => [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[6,4,5,7,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[5,4,6,7,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[8,7,6,5,3,4,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,6,7,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,6,8,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,7,5,8,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[6,5,7,8,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[8,7,6,4,3,5,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,8,6,4,3,5,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,6,8,4,3,5,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[7,8,6,3,4,5,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[8,6,7,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,6,8,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[8,7,5,4,3,6,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St001494
Mp00061: Permutations to increasing treeBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
St001494: Graphs ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0 + 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,5,6,7,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[6,7,5,8,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,5,6,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[6,5,7,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,7,4,5,6,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,6,4,5,7,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[8,4,5,6,7,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[6,7,5,4,8,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[6,5,7,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,6,4,5,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[6,7,4,5,8,3,2,1] => [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [2,1,5,4,3,6,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[7,5,4,6,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[7,4,5,6,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[6,5,4,7,8,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[5,6,4,7,8,3,2,1] => [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [2,1,5,4,3,6,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[6,4,5,7,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[5,4,6,7,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,7,6,5,3,4,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [2,1,3,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,6,7,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,6,8,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [2,1,4,3,6,5,7,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[6,7,5,8,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [2,1,4,3,6,5,7,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 1 + 1
[6,5,7,8,3,4,2,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 + 1
[8,7,6,4,3,5,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,8,6,4,3,5,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [2,1,3,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,6,8,4,3,5,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [1,2,3,6,5,4,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[7,8,6,3,4,5,2,1] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2 + 1
[8,6,7,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,3,2,6,5,4,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[7,6,8,3,4,5,2,1] => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> [1,3,2,6,5,4,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[8,7,5,4,3,6,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1 + 1
Description
The Alon-Tarsi number of a graph. Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St001277
Mp00061: Permutations to increasing treeBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
St001277: Graphs ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 86%
Values
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,6,3,4,5,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,2,7,3,4,5,6] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,3,4,5,6,7,2] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,4,2,3,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,5,2,3,4,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,6,2,3,4,5,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,7,2,3,4,5,6] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[3,1,2,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[4,1,2,3,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[5,1,2,3,4,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[7,8,5,6,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[8,5,6,7,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[6,7,5,8,4,3,2,1] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[7,5,6,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2
[6,5,7,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1
[7,8,6,4,5,3,2,1] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1
[8,7,4,5,6,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[8,6,4,5,7,3,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2
[8,4,5,6,7,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,5,4,3,2,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
Description
The degeneracy of a graph. The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001358The largest degree of a regular subgraph of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St000451The length of the longest pattern of the form k 1 2. St000172The Grundy number of a graph. St000098The chromatic number of a graph. St000527The width of the poset. St000306The bounce count of a Dyck path. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000470The number of runs in a permutation. St000308The height of the tree associated to a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001330The hat guessing number of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001717The largest size of an interval in a poset. St001651The Frankl number of a lattice. St000080The rank of the poset. St001589The nesting number of a perfect matching. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice.